bolzano -weierstrass theorem: meaning, definition, pronunciation and examples

It states that in finite-dimensional Euclidean space (like the real line ℝ or plane ℝ²), any sequence of points that stays within some fixed bounded region must have a subsequence that converges to a limit point within that region.

Bernard Bolzano (1781–1848) was a Czech mathematician and philosopher. Karl Weierstrass (1815–1897) was a German mathematician often called the 'father of modern analysis'. The theorem is named for their contributions to its development and rigorous formulation.

No, it does not. This failure is a key motivation for defining different notions of compactness (like sequential compactness vs. compactness) in functional analysis. In infinite dimensions, bounded sequences may have no convergent subsequences.

The Bolzano-Weierstrass theorem is about sequences (sequential compactness), while the Heine-Borel theorem characterises compact sets in ℝⁿ as those that are closed and bounded. They are logically equivalent in ℝⁿ but offer different perspectives.

A fundamental theorem in real analysis stating that every bounded sequence in ℝⁿ has a convergent subsequence.

Bolzano -weierstrass theorem is usually formal academic/technical in register.

Bolzano -weierstrass theorem: in British English it is pronounced /bɒlˈtsɑːnəʊ ˈvaɪəˌʃtrɑːs ˌθɪərəm/, and in American English it is pronounced /boʊlˈzɑːnoʊ ˈvaɪərˌstræs ˌθɪrəm/. Tap the audio buttons above to hear it.